L(s) = 1 | + (−3.79e4 − 3.79e4i)2-s + (3.04e7 − 3.04e7i)3-s − 1.41e9i·4-s + (7.42e10 − 1.33e11i)5-s − 2.31e12·6-s + (−9.26e12 − 9.26e12i)7-s + (−2.16e14 + 2.16e14i)8-s + 9.78e11i·9-s + (−7.88e15 + 2.24e15i)10-s + 1.98e16·11-s + (−4.29e16 − 4.29e16i)12-s + (−8.93e17 + 8.93e17i)13-s + 7.03e17i·14-s + (−1.79e18 − 6.31e18i)15-s + 1.03e19·16-s + (−3.66e19 − 3.66e19i)17-s + ⋯ |
L(s) = 1 | + (−0.579 − 0.579i)2-s + (0.706 − 0.706i)3-s − 0.328i·4-s + (0.486 − 0.873i)5-s − 0.819·6-s + (−0.278 − 0.278i)7-s + (−0.769 + 0.769i)8-s + 0.000527i·9-s + (−0.788 + 0.224i)10-s + 0.432·11-s + (−0.232 − 0.232i)12-s + (−1.34 + 1.34i)13-s + 0.323i·14-s + (−0.273 − 0.961i)15-s + 0.563·16-s + (−0.753 − 0.753i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0454 - 0.998i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (-0.0454 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{33}{2})\) |
\(\approx\) |
\(0.4696101500\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4696101500\) |
\(L(17)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-7.42e10 + 1.33e11i)T \) |
good | 2 | \( 1 + (3.79e4 + 3.79e4i)T + 4.29e9iT^{2} \) |
| 3 | \( 1 + (-3.04e7 + 3.04e7i)T - 1.85e15iT^{2} \) |
| 7 | \( 1 + (9.26e12 + 9.26e12i)T + 1.10e27iT^{2} \) |
| 11 | \( 1 - 1.98e16T + 2.11e33T^{2} \) |
| 13 | \( 1 + (8.93e17 - 8.93e17i)T - 4.42e35iT^{2} \) |
| 17 | \( 1 + (3.66e19 + 3.66e19i)T + 2.36e39iT^{2} \) |
| 19 | \( 1 + 2.80e20iT - 8.31e40T^{2} \) |
| 23 | \( 1 + (2.06e21 - 2.06e21i)T - 3.76e43iT^{2} \) |
| 29 | \( 1 - 1.02e23iT - 6.26e46T^{2} \) |
| 31 | \( 1 + 9.64e23T + 5.29e47T^{2} \) |
| 37 | \( 1 + (-7.27e24 - 7.27e24i)T + 1.52e50iT^{2} \) |
| 41 | \( 1 - 1.58e25T + 4.06e51T^{2} \) |
| 43 | \( 1 + (9.21e25 - 9.21e25i)T - 1.86e52iT^{2} \) |
| 47 | \( 1 + (3.74e26 + 3.74e26i)T + 3.21e53iT^{2} \) |
| 53 | \( 1 + (-4.43e27 + 4.43e27i)T - 1.50e55iT^{2} \) |
| 59 | \( 1 - 4.27e27iT - 4.64e56T^{2} \) |
| 61 | \( 1 + 5.42e28T + 1.35e57T^{2} \) |
| 67 | \( 1 + (-1.45e29 - 1.45e29i)T + 2.71e58iT^{2} \) |
| 71 | \( 1 + 1.17e29T + 1.73e59T^{2} \) |
| 73 | \( 1 + (8.02e28 - 8.02e28i)T - 4.22e59iT^{2} \) |
| 79 | \( 1 + 2.05e30iT - 5.29e60T^{2} \) |
| 83 | \( 1 + (4.06e30 - 4.06e30i)T - 2.57e61iT^{2} \) |
| 89 | \( 1 + 1.23e31iT - 2.40e62T^{2} \) |
| 97 | \( 1 + (-2.90e31 - 2.90e31i)T + 3.77e63iT^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.44378499498831624130993976672, −13.27859911620536849366686318460, −11.65432421536946885083130403043, −9.671210567413512533799918881743, −8.824253368084103420529460706288, −6.94436424055258988289756763325, −4.89401115117803779520774662122, −2.40009008444455147743097926826, −1.56897917611106422276447025307, −0.14750831446699077080064432824,
2.58262807734597938689910146040, 3.70976761500393408280920809757, 6.15035492681996206929515258446, 7.68779647052863845998996514924, 9.174360082856701450177340534683, 10.21730624924337132877049796568, 12.54988592898841257805497664619, 14.61225815535622945122503493555, 15.42260040369571149221334503298, 17.09789632801578234022045131066